test/geo_transformations.cpp

// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) 2008-2009 Gael Guennebaud <gael.guennebaud@inria.fr>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.

#include "main.h"
#include <Eigen/Geometry>
#include <Eigen/LU>
#include <Eigen/SVD>

template <typename T>
Matrix<T, 2, 1> angleToVec(T a) {
  return Matrix<T, 2, 1>(std::cos(a), std::sin(a));
}

// This permits to workaround a bug in clang/llvm code generation.
template <typename T>
EIGEN_DONT_INLINE void dont_over_optimize(T& x) {
  volatile typename T::Scalar tmp = x(0);
  x(0) = tmp;
}

template <typename Scalar, int Mode, int Options>
void non_projective_only() {
  /* this test covers the following files:
   Cross.h Quaternion.h, Transform.cpp
*/
  typedef Matrix<Scalar, 3, 1> Vector3;
  typedef Quaternion<Scalar> Quaternionx;
  typedef AngleAxis<Scalar> AngleAxisx;
  typedef Transform<Scalar, 3, Mode, Options> Transform3;
  typedef DiagonalMatrix<Scalar, 3> AlignedScaling3;
  typedef Translation<Scalar, 3> Translation3;

  Vector3 v0 = Vector3::Random(), v1 = Vector3::Random();

  Transform3 t0, t1, t2;

  Scalar a = internal::random<Scalar>(-Scalar(EIGEN_PI), Scalar(EIGEN_PI));

  Quaternionx q1, q2;

  q1 = AngleAxisx(a, v0.normalized());

  t0 = Transform3::Identity();
  VERIFY_IS_APPROX(t0.matrix(), Transform3::MatrixType::Identity());

  t0.linear() = q1.toRotationMatrix();

  v0 << 50, 2, 1;
  t0.scale(v0);

  VERIFY_IS_APPROX((t0 * Vector3(1, 0, 0)).template head<3>().norm(), v0.x());

  t0.setIdentity();
  t1.setIdentity();
  v1 << 1, 2, 3;
  t0.linear() = q1.toRotationMatrix();
  t0.pretranslate(v0);
  t0.scale(v1);
  t1.linear() = q1.conjugate().toRotationMatrix();
  t1.prescale(v1.cwiseInverse());
  t1.translate(-v0);

  VERIFY((t0 * t1).matrix().isIdentity(test_precision<Scalar>()));

  t1.fromPositionOrientationScale(v0, q1, v1);
  VERIFY_IS_APPROX(t1.matrix(), t0.matrix());
  VERIFY_IS_APPROX(t1 * v1, t0 * v1);

  // translation * vector
  t0.setIdentity();
  t0.translate(v0);
  VERIFY_IS_APPROX((t0 * v1).template head<3>(), Translation3(v0) * v1);

  // AlignedScaling * vector
  t0.setIdentity();
  t0.scale(v0);
  VERIFY_IS_APPROX((t0 * v1).template head<3>(), AlignedScaling3(v0) * v1);
}

template <typename Scalar, int Mode, int Options>
void transformations() {
  /* this test covers the following files:
     Cross.h Quaternion.h, Transform.cpp
  */
  using std::abs;
  using std::cos;
  typedef Matrix<Scalar, 3, 3> Matrix3;
  typedef Matrix<Scalar, 4, 4> Matrix4;
  typedef Matrix<Scalar, 2, 1> Vector2;
  typedef Matrix<Scalar, 3, 1> Vector3;
  typedef Matrix<Scalar, 4, 1> Vector4;
  typedef Quaternion<Scalar> Quaternionx;
  typedef AngleAxis<Scalar> AngleAxisx;
  typedef Transform<Scalar, 2, Mode, Options> Transform2;
  typedef Transform<Scalar, 3, Mode, Options> Transform3;
  typedef typename Transform3::MatrixType MatrixType;
  typedef DiagonalMatrix<Scalar, 3> AlignedScaling3;
  typedef Translation<Scalar, 2> Translation2;
  typedef Translation<Scalar, 3> Translation3;

  Vector3 v0 = Vector3::Random(), v1 = Vector3::Random();
  Matrix3 matrot1, m;

  Scalar a = internal::random<Scalar>(-Scalar(EIGEN_PI), Scalar(EIGEN_PI));
  Scalar s0 = internal::random<Scalar>(), s1 = internal::random<Scalar>();

  while (v0.norm() < test_precision<Scalar>()) v0 = Vector3::Random();
  while (v1.norm() < test_precision<Scalar>()) v1 = Vector3::Random();

  VERIFY_IS_APPROX(v0, AngleAxisx(a, v0.normalized()) * v0);
  VERIFY_IS_APPROX(-v0, AngleAxisx(Scalar(EIGEN_PI), v0.unitOrthogonal()) * v0);
  if (abs(cos(a)) > test_precision<Scalar>()) {
    VERIFY_IS_APPROX(cos(a) * v0.squaredNorm(), v0.dot(AngleAxisx(a, v0.unitOrthogonal()) * v0));
  }
  m = AngleAxisx(a, v0.normalized()).toRotationMatrix().adjoint();
  VERIFY_IS_APPROX(Matrix3::Identity(), m * AngleAxisx(a, v0.normalized()));
  VERIFY_IS_APPROX(Matrix3::Identity(), AngleAxisx(a, v0.normalized()) * m);

  Quaternionx q1, q2;
  q1 = AngleAxisx(a, v0.normalized());
  q2 = AngleAxisx(a, v1.normalized());

  // rotation matrix conversion
  matrot1 = AngleAxisx(Scalar(0.1), Vector3::UnitX()) * AngleAxisx(Scalar(0.2), Vector3::UnitY()) *
            AngleAxisx(Scalar(0.3), Vector3::UnitZ());
  VERIFY_IS_APPROX(matrot1 * v1, AngleAxisx(Scalar(0.1), Vector3(1, 0, 0)).toRotationMatrix() *
                                     (AngleAxisx(Scalar(0.2), Vector3(0, 1, 0)).toRotationMatrix() *
                                      (AngleAxisx(Scalar(0.3), Vector3(0, 0, 1)).toRotationMatrix() * v1)));

  // angle-axis conversion
  AngleAxisx aa = AngleAxisx(q1);
  VERIFY_IS_APPROX(q1 * v1, Quaternionx(aa) * v1);

  // The following test is stable only if 2*angle != angle and v1 is not colinear with axis
  if ((abs(aa.angle()) > test_precision<Scalar>()) &&
      (abs(aa.axis().dot(v1.normalized())) < (Scalar(1) - Scalar(4) * test_precision<Scalar>()))) {
    VERIFY(!(q1 * v1).isApprox(Quaternionx(AngleAxisx(aa.angle() * 2, aa.axis())) * v1));
  }

  aa.fromRotationMatrix(aa.toRotationMatrix());
  VERIFY_IS_APPROX(q1 * v1, Quaternionx(aa) * v1);
  // The following test is stable only if 2*angle != angle and v1 is not colinear with axis
  if ((abs(aa.angle()) > test_precision<Scalar>()) &&
      (abs(aa.axis().dot(v1.normalized())) < (Scalar(1) - Scalar(4) * test_precision<Scalar>()))) {
    VERIFY(!(q1 * v1).isApprox(Quaternionx(AngleAxisx(aa.angle() * 2, aa.axis())) * v1));
  }

  // AngleAxis
  VERIFY_IS_APPROX(AngleAxisx(a, v1.normalized()).toRotationMatrix(),
                   Quaternionx(AngleAxisx(a, v1.normalized())).toRotationMatrix());

  AngleAxisx aa1;
  m = q1.toRotationMatrix();
  aa1 = m;
  VERIFY_IS_APPROX(AngleAxisx(m).toRotationMatrix(), Quaternionx(m).toRotationMatrix());

  // Transform
  // TODO complete the tests !
  a = 0;
  while (abs(a) < Scalar(0.1))
    a = internal::random<Scalar>(-Scalar(0.4) * Scalar(EIGEN_PI), Scalar(0.4) * Scalar(EIGEN_PI));
  q1 = AngleAxisx(a, v0.normalized());
  Transform3 t0, t1, t2;

  // first test setIdentity() and Identity()
  t0.setIdentity();
  VERIFY_IS_APPROX(t0.matrix(), Transform3::MatrixType::Identity());
  t0.matrix().setZero();
  t0 = Transform3::Identity();
  VERIFY_IS_APPROX(t0.matrix(), Transform3::MatrixType::Identity());

  t0.setIdentity();
  t1.setIdentity();
  v1 << 1, 2, 3;
  t0.linear() = q1.toRotationMatrix();
  t0.pretranslate(v0);
  t0.scale(v1);
  t1.linear() = q1.conjugate().toRotationMatrix();
  t1.prescale(v1.cwiseInverse());
  t1.translate(-v0);

  VERIFY((t0 * t1).matrix().isIdentity(test_precision<Scalar>()));

  t1.fromPositionOrientationScale(v0, q1, v1);
  VERIFY_IS_APPROX(t1.matrix(), t0.matrix());

  t0.setIdentity();
  t0.scale(v0).rotate(q1.toRotationMatrix());
  t1.setIdentity();
  t1.scale(v0).rotate(q1);
  VERIFY_IS_APPROX(t0.matrix(), t1.matrix());

  t0.setIdentity();
  t0.scale(v0).rotate(AngleAxisx(q1));
  VERIFY_IS_APPROX(t0.matrix(), t1.matrix());

  VERIFY_IS_APPROX(t0.scale(a).matrix(), t1.scale(Vector3::Constant(a)).matrix());
  VERIFY_IS_APPROX(t0.prescale(a).matrix(), t1.prescale(Vector3::Constant(a)).matrix());

  // More transform constructors, operator=, operator*=

  Matrix3 mat3 = Matrix3::Random();
  Matrix4 mat4;
  mat4 << mat3, Vector3::Zero(), Vector4::Zero().transpose();
  Transform3 tmat3(mat3), tmat4(mat4);
  if (Mode != int(AffineCompact)) tmat4.matrix()(3, 3) = Scalar(1);
  VERIFY_IS_APPROX(tmat3.matrix(), tmat4.matrix());

  Scalar a3 = internal::random<Scalar>(-Scalar(EIGEN_PI), Scalar(EIGEN_PI));
  Vector3 v3 = Vector3::Random().normalized();
  AngleAxisx aa3(a3, v3);
  Transform3 t3(aa3);
  Transform3 t4;
  t4 = aa3;
  VERIFY_IS_APPROX(t3.matrix(), t4.matrix());
  t4.rotate(AngleAxisx(-a3, v3));
  VERIFY_IS_APPROX(t4.matrix(), MatrixType::Identity());
  t4 *= aa3;
  VERIFY_IS_APPROX(t3.matrix(), t4.matrix());

  do {
    v3 = Vector3::Random();
    dont_over_optimize(v3);
  } while (v3.cwiseAbs().minCoeff() < NumTraits<Scalar>::epsilon());
  Translation3 tv3(v3);
  Transform3 t5(tv3);
  t4 = tv3;
  VERIFY_IS_APPROX(t5.matrix(), t4.matrix());
  t4.translate((-v3).eval());
  VERIFY_IS_APPROX(t4.matrix(), MatrixType::Identity());
  t4 *= tv3;
  VERIFY_IS_APPROX(t5.matrix(), t4.matrix());

  AlignedScaling3 sv3(v3);
  Transform3 t6(sv3);
  t4 = sv3;
  VERIFY_IS_APPROX(t6.matrix(), t4.matrix());
  t4.scale(v3.cwiseInverse());
  VERIFY_IS_APPROX(t4.matrix(), MatrixType::Identity());
  t4 *= sv3;
  VERIFY_IS_APPROX(t6.matrix(), t4.matrix());

  // matrix * transform
  VERIFY_IS_APPROX((t3.matrix() * t4).matrix(), (t3 * t4).matrix());

  // chained Transform product
  VERIFY_IS_APPROX(((t3 * t4) * t5).matrix(), (t3 * (t4 * t5)).matrix());

  // check that Transform product doesn't have aliasing problems
  t5 = t4;
  t5 = t5 * t5;
  VERIFY_IS_APPROX(t5, t4 * t4);

  // 2D transformation
  Transform2 t20, t21;
  Vector2 v20 = Vector2::Random();
  Vector2 v21 = Vector2::Random();
  for (int k = 0; k < 2; ++k)
    if (abs(v21[k]) < Scalar(1e-3)) v21[k] = Scalar(1e-3);
  t21.setIdentity();
  t21.linear() = Rotation2D<Scalar>(a).toRotationMatrix();
  VERIFY_IS_APPROX(t20.fromPositionOrientationScale(v20, a, v21).matrix(), t21.pretranslate(v20).scale(v21).matrix());

  t21.setIdentity();
  t21.linear() = Rotation2D<Scalar>(-a).toRotationMatrix();
  VERIFY((t20.fromPositionOrientationScale(v20, a, v21) * (t21.prescale(v21.cwiseInverse()).translate(-v20)))
             .matrix()
             .isIdentity(test_precision<Scalar>()));

  // Transform - new API
  // 3D
  t0.setIdentity();
  t0.rotate(q1).scale(v0).translate(v0);
  // mat * aligned scaling and mat * translation
  t1 = (Matrix3(q1) * AlignedScaling3(v0)) * Translation3(v0);
  VERIFY_IS_APPROX(t0.matrix(), t1.matrix());
  t1 = (Matrix3(q1) * Eigen::Scaling(v0)) * Translation3(v0);
  VERIFY_IS_APPROX(t0.matrix(), t1.matrix());
  t1 = (q1 * Eigen::Scaling(v0)) * Translation3(v0);
  VERIFY_IS_APPROX(t0.matrix(), t1.matrix());
  // mat * transformation and aligned scaling * translation
  t1 = Matrix3(q1) * (AlignedScaling3(v0) * Translation3(v0));
  VERIFY_IS_APPROX(t0.matrix(), t1.matrix());

  t0.setIdentity();
  t0.scale(s0).translate(v0);
  t1 = Eigen::Scaling(s0) * Translation3(v0);
  VERIFY_IS_APPROX(t0.matrix(), t1.matrix());
  t0.prescale(s0);
  t1 = Eigen::Scaling(s0) * t1;
  VERIFY_IS_APPROX(t0.matrix(), t1.matrix());

  t0 = t3;
  t0.scale(s0);
  t1 = t3 * Eigen::Scaling(s0, s0, s0);
  VERIFY_IS_APPROX(t0.matrix(), t1.matrix());
  t0.prescale(s0);
  t1 = Eigen::Scaling(s0, s0, s0) * t1;
  VERIFY_IS_APPROX(t0.matrix(), t1.matrix());

  t0 = t3;
  t0.scale(s0);
  t1 = t3 * Eigen::Scaling(s0);
  VERIFY_IS_APPROX(t0.matrix(), t1.matrix());
  t0.prescale(s0);
  t1 = Eigen::Scaling(s0) * t1;
  VERIFY_IS_APPROX(t0.matrix(), t1.matrix());

  t0.setIdentity();
  t0.prerotate(q1).prescale(v0).pretranslate(v0);
  // translation * aligned scaling and transformation * mat
  t1 = (Translation3(v0) * AlignedScaling3(v0)) * Transform3(q1);
  VERIFY_IS_APPROX(t0.matrix(), t1.matrix());
  // scaling * mat and translation * mat
  t1 = Translation3(v0) * (AlignedScaling3(v0) * Transform3(q1));
  VERIFY_IS_APPROX(t0.matrix(), t1.matrix());

  t0.setIdentity();
  t0.scale(v0).translate(v0).rotate(q1);
  // translation * mat and aligned scaling * transformation
  t1 = AlignedScaling3(v0) * (Translation3(v0) * Transform3(q1));
  VERIFY_IS_APPROX(t0.matrix(), t1.matrix());
  // transformation * aligned scaling
  t0.scale(v0);
  t1 *= AlignedScaling3(v0);
  VERIFY_IS_APPROX(t0.matrix(), t1.matrix());
  t1 = AlignedScaling3(v0) * (Translation3(v0) * Transform3(q1));
  t1 = t1 * v0.asDiagonal();
  VERIFY_IS_APPROX(t0.matrix(), t1.matrix());
  // transformation * translation
  t0.translate(v0);
  t1 = t1 * Translation3(v0);
  VERIFY_IS_APPROX(t0.matrix(), t1.matrix());
  // translation * transformation
  t0.pretranslate(v0);
  t1 = Translation3(v0) * t1;
  VERIFY_IS_APPROX(t0.matrix(), t1.matrix());

  // transform * quaternion
  t0.rotate(q1);
  t1 = t1 * q1;
  VERIFY_IS_APPROX(t0.matrix(), t1.matrix());

  // translation * quaternion
  t0.translate(v1).rotate(q1);
  t1 = t1 * (Translation3(v1) * q1);
  VERIFY_IS_APPROX(t0.matrix(), t1.matrix());

  // aligned scaling * quaternion
  t0.scale(v1).rotate(q1);
  t1 = t1 * (AlignedScaling3(v1) * q1);
  VERIFY_IS_APPROX(t0.matrix(), t1.matrix());

  // quaternion * transform
  t0.prerotate(q1);
  t1 = q1 * t1;
  VERIFY_IS_APPROX(t0.matrix(), t1.matrix());

  // quaternion * translation
  t0.rotate(q1).translate(v1);
  t1 = t1 * (q1 * Translation3(v1));
  VERIFY_IS_APPROX(t0.matrix(), t1.matrix());

  // quaternion * aligned scaling
  t0.rotate(q1).scale(v1);
  t1 = t1 * (q1 * AlignedScaling3(v1));
  VERIFY_IS_APPROX(t0.matrix(), t1.matrix());

  // test transform inversion
  t0.setIdentity();
  t0.translate(v0);
  do {
    t0.linear().setRandom();
  } while (t0.linear().jacobiSvd().singularValues()(2) < test_precision<Scalar>());
  Matrix4 t044 = Matrix4::Zero();
  t044(3, 3) = 1;
  t044.block(0, 0, t0.matrix().rows(), 4) = t0.matrix();
  VERIFY_IS_APPROX(t0.inverse(Affine).matrix(), t044.inverse().block(0, 0, t0.matrix().rows(), 4));
  t0.setIdentity();
  t0.translate(v0).rotate(q1);
  t044 = Matrix4::Zero();
  t044(3, 3) = 1;
  t044.block(0, 0, t0.matrix().rows(), 4) = t0.matrix();
  VERIFY_IS_APPROX(t0.inverse(Isometry).matrix(), t044.inverse().block(0, 0, t0.matrix().rows(), 4));

  Matrix3 mat_rotation, mat_scaling;
  t0.setIdentity();
  t0.translate(v0).rotate(q1).scale(v1);
  t0.computeRotationScaling(&mat_rotation, &mat_scaling);
  VERIFY_IS_APPROX(t0.linear(), mat_rotation * mat_scaling);
  VERIFY_IS_APPROX(mat_rotation * mat_rotation.adjoint(), Matrix3::Identity());
  VERIFY_IS_APPROX(mat_rotation.determinant(), Scalar(1));
  t0.computeScalingRotation(&mat_scaling, &mat_rotation);
  VERIFY_IS_APPROX(t0.linear(), mat_scaling * mat_rotation);
  VERIFY_IS_APPROX(mat_rotation * mat_rotation.adjoint(), Matrix3::Identity());
  VERIFY_IS_APPROX(mat_rotation.determinant(), Scalar(1));

  // test casting
  Transform<float, 3, Mode> t1f = t1.template cast<float>();
  VERIFY_IS_APPROX(t1f.template cast<Scalar>(), t1);
  Transform<double, 3, Mode> t1d = t1.template cast<double>();
  VERIFY_IS_APPROX(t1d.template cast<Scalar>(), t1);

  Translation3 tr1(v0);
  Translation<float, 3> tr1f = tr1.template cast<float>();
  VERIFY_IS_APPROX(tr1f.template cast<Scalar>(), tr1);
  Translation<double, 3> tr1d = tr1.template cast<double>();
  VERIFY_IS_APPROX(tr1d.template cast<Scalar>(), tr1);

  AngleAxis<float> aa1f = aa1.template cast<float>();
  VERIFY_IS_APPROX(aa1f.template cast<Scalar>(), aa1);
  AngleAxis<double> aa1d = aa1.template cast<double>();
  VERIFY_IS_APPROX(aa1d.template cast<Scalar>(), aa1);

  Rotation2D<Scalar> r2d1(internal::random<Scalar>());
  Rotation2D<float> r2d1f = r2d1.template cast<float>();
  VERIFY_IS_APPROX(r2d1f.template cast<Scalar>(), r2d1);
  Rotation2D<double> r2d1d = r2d1.template cast<double>();
  VERIFY_IS_APPROX(r2d1d.template cast<Scalar>(), r2d1);

  for (int k = 0; k < 100; ++k) {
    Scalar angle = internal::random<Scalar>(-100, 100);
    Rotation2D<Scalar> rot2(angle);
    VERIFY(rot2.smallestPositiveAngle() >= 0);
    VERIFY(rot2.smallestPositiveAngle() <= Scalar(2) * Scalar(EIGEN_PI));
    VERIFY_IS_APPROX(angleToVec(rot2.smallestPositiveAngle()), angleToVec(rot2.angle()));

    VERIFY(rot2.smallestAngle() >= -Scalar(EIGEN_PI));
    VERIFY(rot2.smallestAngle() <= Scalar(EIGEN_PI));
    VERIFY_IS_APPROX(angleToVec(rot2.smallestAngle()), angleToVec(rot2.angle()));

    Matrix<Scalar, 2, 2> rot2_as_mat(rot2);
    Rotation2D<Scalar> rot3(rot2_as_mat);
    VERIFY_IS_APPROX(angleToVec(rot2.smallestAngle()), angleToVec(rot3.angle()));
  }

  s0 = internal::random<Scalar>(-100, 100);
  s1 = internal::random<Scalar>(-100, 100);
  Rotation2D<Scalar> R0(s0), R1(s1);

  t20 = Translation2(v20) * (R0 * Eigen::Scaling(s0));
  t21 = Translation2(v20) * R0 * Eigen::Scaling(s0);
  VERIFY_IS_APPROX(t20, t21);

  t20 = Translation2(v20) * (R0 * R0.inverse() * Eigen::Scaling(s0));
  t21 = Translation2(v20) * Eigen::Scaling(s0);
  VERIFY_IS_APPROX(t20, t21);

  VERIFY_IS_APPROX(s0, (R0.slerp(0, R1)).angle());
  VERIFY_IS_APPROX(angleToVec(R1.smallestPositiveAngle()), angleToVec((R0.slerp(1, R1)).smallestPositiveAngle()));
  VERIFY_IS_APPROX(R0.smallestPositiveAngle(), (R0.slerp(0.5, R0)).smallestPositiveAngle());

  if (std::cos(s0) > 0)
    VERIFY_IS_MUCH_SMALLER_THAN((R0.slerp(0.5, R0.inverse())).smallestAngle(), Scalar(1));
  else
    VERIFY_IS_APPROX(Scalar(EIGEN_PI), (R0.slerp(0.5, R0.inverse())).smallestPositiveAngle());

  // Check path length
  Scalar l = 0;
  int path_steps = 100;
  for (int k = 0; k < path_steps; ++k) {
    Scalar a1 = R0.slerp(Scalar(k) / Scalar(path_steps), R1).angle();
    Scalar a2 = R0.slerp(Scalar(k + 1) / Scalar(path_steps), R1).angle();
    l += std::abs(a2 - a1);
  }
  VERIFY(l <= Scalar(EIGEN_PI) * (Scalar(1) + NumTraits<Scalar>::epsilon() * Scalar(path_steps / 2)));

  // check basic features
  {
    Rotation2D<Scalar> r1;        // default ctor
    r1 = Rotation2D<Scalar>(s0);  // copy assignment
    VERIFY_IS_APPROX(r1.angle(), s0);
    Rotation2D<Scalar> r2(r1);  // copy ctor
    VERIFY_IS_APPROX(r2.angle(), s0);
  }

  {
    Transform3 t32(Matrix4::Random()), t33, t34;
    t34 = t33 = t32;
    t32.scale(v0);
    t33 *= AlignedScaling3(v0);
    VERIFY_IS_APPROX(t32.matrix(), t33.matrix());
    t33 = t34 * AlignedScaling3(v0);
    VERIFY_IS_APPROX(t32.matrix(), t33.matrix());
  }
}

template <typename A1, typename A2, typename P, typename Q, typename V, typename H>
void transform_associativity_left(const A1& a1, const A2& a2, const P& p, const Q& q, const V& v, const H& h) {
  VERIFY_IS_APPROX(q * (a1 * v), (q * a1) * v);
  VERIFY_IS_APPROX(q * (a2 * v), (q * a2) * v);
  VERIFY_IS_APPROX(q * (p * h).hnormalized(), ((q * p) * h).hnormalized());
}

template <typename A1, typename A2, typename P, typename Q, typename V, typename H>
void transform_associativity2(const A1& a1, const A2& a2, const P& p, const Q& q, const V& v, const H& h) {
  VERIFY_IS_APPROX(a1 * (q * v), (a1 * q) * v);
  VERIFY_IS_APPROX(a2 * (q * v), (a2 * q) * v);
  VERIFY_IS_APPROX(p * (q * v).homogeneous(), (p * q) * v.homogeneous());

  transform_associativity_left(a1, a2, p, q, v, h);
}

template <typename Scalar, int Dim, int Options, typename RotationType>
void transform_associativity(const RotationType& R) {
  typedef Matrix<Scalar, Dim, 1> VectorType;
  typedef Matrix<Scalar, Dim + 1, 1> HVectorType;
  typedef Matrix<Scalar, Dim, Dim> LinearType;
  typedef Matrix<Scalar, Dim + 1, Dim + 1> MatrixType;
  typedef Transform<Scalar, Dim, AffineCompact, Options> AffineCompactType;
  typedef Transform<Scalar, Dim, Affine, Options> AffineType;
  typedef Transform<Scalar, Dim, Projective, Options> ProjectiveType;
  typedef DiagonalMatrix<Scalar, Dim> ScalingType;
  typedef Translation<Scalar, Dim> TranslationType;

  AffineCompactType A1c;
  A1c.matrix().setRandom();
  AffineCompactType A2c;
  A2c.matrix().setRandom();
  AffineType A1(A1c);
  AffineType A2(A2c);
  ProjectiveType P1;
  P1.matrix().setRandom();
  VectorType v1 = VectorType::Random();
  VectorType v2 = VectorType::Random();
  HVectorType h1 = HVectorType::Random();
  Scalar s1 = internal::random<Scalar>();
  LinearType L = LinearType::Random();
  MatrixType M = MatrixType::Random();

  CALL_SUBTEST(transform_associativity2(A1c, A1, P1, A2, v2, h1));
  CALL_SUBTEST(transform_associativity2(A1c, A1, P1, A2c, v2, h1));
  CALL_SUBTEST(transform_associativity2(A1c, A1, P1, v1.asDiagonal(), v2, h1));
  CALL_SUBTEST(transform_associativity2(A1c, A1, P1, ScalingType(v1), v2, h1));
  CALL_SUBTEST(transform_associativity2(A1c, A1, P1, Scaling(v1), v2, h1));
  CALL_SUBTEST(transform_associativity2(A1c, A1, P1, Scaling(s1), v2, h1));
  CALL_SUBTEST(transform_associativity2(A1c, A1, P1, TranslationType(v1), v2, h1));
  CALL_SUBTEST(transform_associativity_left(A1c, A1, P1, L, v2, h1));
  CALL_SUBTEST(transform_associativity2(A1c, A1, P1, R, v2, h1));

  VERIFY_IS_APPROX(A1 * (M * h1), (A1 * M) * h1);
  VERIFY_IS_APPROX(A1c * (M * h1), (A1c * M) * h1);
  VERIFY_IS_APPROX(P1 * (M * h1), (P1 * M) * h1);

  VERIFY_IS_APPROX(M * (A1 * h1), (M * A1) * h1);
  VERIFY_IS_APPROX(M * (A1c * h1), (M * A1c) * h1);
  VERIFY_IS_APPROX(M * (P1 * h1), ((M * P1) * h1));
}

template <typename Scalar>
void transform_alignment() {
  typedef Transform<Scalar, 3, Projective, AutoAlign> Projective3a;
  typedef Transform<Scalar, 3, Projective, DontAlign> Projective3u;

  EIGEN_ALIGN_MAX Scalar array1[16];
  EIGEN_ALIGN_MAX Scalar array2[16];
  EIGEN_ALIGN_MAX Scalar array3[16 + 1];
  Scalar* array3u = array3 + 1;

  Projective3a* p1 = ::new (reinterpret_cast<void*>(array1)) Projective3a;
  Projective3u* p2 = ::new (reinterpret_cast<void*>(array2)) Projective3u;
  Projective3u* p3 = ::new (reinterpret_cast<void*>(array3u)) Projective3u;

  p1->matrix().setRandom();
  *p2 = *p1;
  *p3 = *p1;

  VERIFY_IS_APPROX(p1->matrix(), p2->matrix());
  VERIFY_IS_APPROX(p1->matrix(), p3->matrix());

  VERIFY_IS_APPROX((*p1) * (*p1), (*p2) * (*p3));
}

template <typename Scalar, int Dim, int Options>
void transform_products() {
  typedef Matrix<Scalar, Dim + 1, Dim + 1> Mat;
  typedef Transform<Scalar, Dim, Projective, Options> Proj;
  typedef Transform<Scalar, Dim, Affine, Options> Aff;
  typedef Transform<Scalar, Dim, AffineCompact, Options> AffC;

  Proj p;
  p.matrix().setRandom();
  Aff a;
  a.linear().setRandom();
  a.translation().setRandom();
  AffC ac = a;

  Mat p_m(p.matrix()), a_m(a.matrix());

  VERIFY_IS_APPROX((p * p).matrix(), p_m * p_m);
  VERIFY_IS_APPROX((a * a).matrix(), a_m * a_m);
  VERIFY_IS_APPROX((p * a).matrix(), p_m * a_m);
  VERIFY_IS_APPROX((a * p).matrix(), a_m * p_m);
  VERIFY_IS_APPROX((ac * a).matrix(), a_m * a_m);
  VERIFY_IS_APPROX((a * ac).matrix(), a_m * a_m);
  VERIFY_IS_APPROX((p * ac).matrix(), p_m * a_m);
  VERIFY_IS_APPROX((ac * p).matrix(), a_m * p_m);
}

template <typename Scalar, int Mode, int Options>
void transformations_no_scale() {
  /* this test covers the following files:
  Cross.h Quaternion.h, Transform.h
*/
  typedef Matrix<Scalar, 3, 1> Vector3;
  typedef Matrix<Scalar, 4, 1> Vector4;
  typedef Quaternion<Scalar> Quaternionx;
  typedef AngleAxis<Scalar> AngleAxisx;
  typedef Transform<Scalar, 3, Mode, Options> Transform3;
  typedef Translation<Scalar, 3> Translation3;
  typedef Matrix<Scalar, 4, 4> Matrix4;

  Vector3 v0 = Vector3::Random(), v1 = Vector3::Random();

  Transform3 t0, t1, t2;

  Scalar a = internal::random<Scalar>(-Scalar(EIGEN_PI), Scalar(EIGEN_PI));

  Quaternionx q1, q2;

  q1 = AngleAxisx(a, v0.normalized());

  t0 = Transform3::Identity();
  VERIFY_IS_APPROX(t0.matrix(), Transform3::MatrixType::Identity());

  t0.setIdentity();
  t1.setIdentity();
  v1 = Vector3::Ones();
  t0.linear() = q1.toRotationMatrix();
  t0.pretranslate(v0);
  t1.linear() = q1.conjugate().toRotationMatrix();
  t1.translate(-v0);

  VERIFY((t0 * t1).matrix().isIdentity(test_precision<Scalar>()));

  t1.fromPositionOrientationScale(v0, q1, v1);
  VERIFY_IS_APPROX(t1.matrix(), t0.matrix());
  VERIFY_IS_APPROX(t1 * v1, t0 * v1);

  // translation * vector
  t0.setIdentity();
  t0.translate(v0);
  VERIFY_IS_APPROX((t0 * v1).template head<3>(), Translation3(v0) * v1);

  // Conversion to matrix.
  Transform3 t3;
  t3.linear() = q1.toRotationMatrix();
  t3.translation() = v1;
  Matrix4 m3 = t3.matrix();
  VERIFY((m3 * m3.inverse()).isIdentity(test_precision<Scalar>()));
  // Verify implicit last row is initialized.
  VERIFY_IS_APPROX(Vector4(m3.row(3)), Vector4(0.0, 0.0, 0.0, 1.0));

  VERIFY_IS_APPROX(t3.rotation(), t3.linear());
  if (Mode == Isometry) VERIFY(t3.rotation().data() == t3.linear().data());
}

template <typename Scalar, int Mode, int Options>
void transformations_computed_scaling_continuity() {
  typedef Matrix<Scalar, 3, 1> Vector3;
  typedef Transform<Scalar, 3, Mode, Options> Transform3;
  typedef Matrix<Scalar, 3, 3> Matrix3;

  // Given: two transforms that differ by '2*eps'.
  Scalar eps(1e-3);
  Vector3 v0 = Vector3::Random().normalized(), v1 = Vector3::Random().normalized(), v3 = Vector3::Random().normalized();
  Transform3 t0, t1;
  // The interesting case is when their determinants have different signs.
  Matrix3 rank2 = 50 * v0 * v0.adjoint() + 20 * v1 * v1.adjoint();
  t0.linear() = rank2 + eps * v3 * v3.adjoint();
  t1.linear() = rank2 - eps * v3 * v3.adjoint();

  // When: computing the rotation-scaling parts
  Matrix3 r0, s0, r1, s1;
  t0.computeRotationScaling(&r0, &s0);
  t1.computeRotationScaling(&r1, &s1);

  // Then: the scaling parts should differ by no more than '2*eps'.
  const Scalar c(2.1);  // 2 + room for rounding errors
  VERIFY((s0 - s1).norm() < c * eps);
}

EIGEN_DECLARE_TEST(geo_transformations) {
  for (int i = 0; i < g_repeat; i++) {
    CALL_SUBTEST_1((transformations<double, Affine, AutoAlign>()));
    CALL_SUBTEST_1((non_projective_only<double, Affine, AutoAlign>()));
    CALL_SUBTEST_1((transformations_computed_scaling_continuity<double, Affine, AutoAlign>()));

    CALL_SUBTEST_2((transformations<float, AffineCompact, AutoAlign>()));
    CALL_SUBTEST_2((non_projective_only<float, AffineCompact, AutoAlign>()));
    CALL_SUBTEST_2((transform_alignment<float>()));

    CALL_SUBTEST_3((transformations<double, Projective, AutoAlign>()));
    CALL_SUBTEST_3((transformations<double, Projective, DontAlign>()));
    CALL_SUBTEST_3((transform_alignment<double>()));

    CALL_SUBTEST_4((transformations<float, Affine, RowMajor | AutoAlign>()));
    CALL_SUBTEST_4((non_projective_only<float, Affine, RowMajor>()));

    CALL_SUBTEST_5((transformations<double, AffineCompact, RowMajor | AutoAlign>()));
    CALL_SUBTEST_5((non_projective_only<double, AffineCompact, RowMajor>()));

    CALL_SUBTEST_6((transformations<double, Projective, RowMajor | AutoAlign>()));
    CALL_SUBTEST_6((transformations<double, Projective, RowMajor | DontAlign>()));

    CALL_SUBTEST_7((transform_products<double, 3, RowMajor | AutoAlign>()));
    CALL_SUBTEST_7((transform_products<float, 2, AutoAlign>()));

    CALL_SUBTEST_8((transform_associativity<double, 2, ColMajor>(
        Rotation2D<double>(internal::random<double>() * double(EIGEN_PI)))));
    CALL_SUBTEST_8((transform_associativity<double, 3, ColMajor>(Quaterniond::UnitRandom())));

    CALL_SUBTEST_9((transformations_no_scale<double, Affine, AutoAlign>()));
    CALL_SUBTEST_9((transformations_no_scale<double, Isometry, AutoAlign>()));
  }
}